MOTION DIAGRAM 2
There is another type of motion diagram that is, in many ways, more versatile than the original
type of motion diagram. This new type of motion diagram, which we call motion diagram 2, can
be used for motion in more than one-dimension and for linear motion. This motion diagram
2 is sometimes called “finding acceleration from known velocities”.
We know that a motion diagram can represent the position, velocity, and acceleration of an
object at several different times. In this type of motion diagram, you will be able to find the
direction of an object’s acceleration at some selected time t. The direction of the acceleration
can be determined if we know the objects velocity at two different times separated by a short
time interval t. We will drawing motion diagrams of constant acceleration or regions of
constant acceleration.
Procedure:
To draw a motion diagram using this
technique, you want to first sketch the path
of the object’s motion. On the object’s path,
choose the point for some time t at which
you want to determine the direction of the
acceleration.
Now, choose a point a little before the
selected point of time t. This will be known
as the original velocity or initial velocity.
At this earlier point, draw an arrow to
represent the velocity of the object at this
point. Note: that the velocity will be tangent
to the path at this point. We will call this the
original velocity and label it vo.
Now, choose a point a little after the
selected point of time t. This will be known as
the later velocity or final velocity. At this
later point, draw an arrow to represent the
velocity of the object at this point. Note that
the velocity will be tangent to the path at this
point. We will call this the final velocity and
label it vf.
Revised 8/06
-1-
t
vo
t
vf
vo
t
©LC, tlo
Now, you need to find the change in
velocity for the object between the initial
velocity and the final velocity. This is
known as the change in velocity and will
represent the velocity change for the object
at time t. To find the change in velocity v
during the time interval t, place the tails of
the vo and vf together. The change if
velocity is a vector that points from the head
of vo to the head of vf. Notice in the figure
that vo + v = vf or rearranging, v = vf – vo
(that is, v is the change in velocity).
v
vo
t
vf
Finally, the direction of the acceleration is always in the same direction as the velocity
change. The acceleration equals the velocity change v divided by the time interval t needed
for that change; that is, a = v/t. If you do not know the time interval, you can at least
determine the direction of the acceleration because it points in the same direction as v.
t
a
v
t
In summary, notice that the direction of the acceleration is towards the “inside” of the
trajectory.
Revised 8/08
-2-
©LC, tlo
Motion Diagram 2 - 1
A ball moving at a constant speed rolls off
the edge of a horizontal table. Determine
the direction of the ball's acceleration when
at a position after it left the table, but before
it hits the floor. The velocity at each point
along the path of the ball is tangent to the
path.
Earlier Velocity: Draw an arrow
representing the velocity vo of the ball
at time to when at a position just before
the desired point on the path.
Later Velocity: Draw another arrow
representing the velocity vf of the ball
at a later time tf when at a position just
after the desired point on the path.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-3-
©LC, tlo
Motion Diagram 2 - 2
A ball is thrown with an initial velocity at
an angle above the horizontal. Determine
the direction of the ball's acceleration when
at a position after it leaves your hand, but
before it hits the floor. The velocity at each
point along the path of the ball is tangent to
the path.
Earlier Velocity: Draw an arrow
representing the velocity vo of the ball
at time to when at a position just before
the desired point on the path.
Later Velocity: Draw another arrow
representing the velocity vf of the ball
at a later time tf when at a position just
after the desired point on the path.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-4-
©LC, tlo
Motion Diagram 2 – 3a
A car heading north on a road makes a right
turn onto a side street. Determine the
direction of the car's acceleration when
at a position during the turn. The velocity
at each point along the path of the car is
tangent to the path of the car if viewed from
above.
Earlier Velocity: Draw an arrow
representing the velocity vo of the car
at time to when at a position just before
the desired point on the turn.
Later Velocity: Draw another arrow
representing the velocity vf of the car
at a later time tf when at a position just
after the desired point on the turn.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-5-
©LC, tlo
Motion Diagram 2 – 3b
A car heading north on a road makes a left
turn onto a side street. Determine the
direction of the car's acceleration when
at a position during the turn. The velocity
at each point along the path of the car is
tangent to the path of the car if viewed from
above.
Earlier Velocity: Draw an arrow
representing the velocity vo of the car
at time to when at a position just before
the desired point on the turn.
Later Velocity: Draw another arrow
representing the velocity vf of the car
at a later time tf when at a position just
after the desired point on the turn.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-6-
©LC, tlo
Motion Diagram 2 – 3c
A car heading south on a road makes a left
turn onto a side street. Determine the
direction of the car's acceleration when
at a position during the turn. The velocity
at each point along the path of the car is
tangent to the path of the car if viewed from
above.
Earlier Velocity: Draw an arrow
representing the velocity vo of the car
at time to when at a position just before
the desired point on the turn.
Later Velocity: Draw another arrow
representing the velocity vf of the car
at a later time tf when at a position just
after the desired point on the turn.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-7-
©LC, tlo
Motion Diagram 2 – 3d
A car heading east on a road makes a left
turn onto a side street. Determine the
direction of the car's acceleration when
at a position during the turn. The velocity
at each point along the path of the car is
tangent to the path of the car if viewed from
above.
Earlier Velocity: Draw an arrow
representing the velocity vo of the car
at time to when at a position just before
the desired point on the turn.
Later Velocity: Draw another arrow
representing the velocity vf of the car
at a later time tf when at a position just
after the desired point on the turn.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-8-
©LC, tlo
Motion Diagram 2 – 4a
An object starting from rest rolls down a slide
as indicated in the figure. Determine the
direction of the object's acceleration at a point
along the slide before it reaches the bottom.
The velocity at each point along the slide is
tangent to the slide. The speed of the object
increases as it rolls down the slide.
Earlier Velocity: Draw an arrow
representing the velocity vo of the object
at time to when at a position just before
the desired point on the slide.
Later Velocity: Draw another arrow
representing the velocity vf of the object
at a later time tf when at a position just
after the desired point on the slide.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
-9-
©LC, tlo
Motion Diagram 2 – 4b
An object moving at some velocity rolls up a
slide as indicated in the figure. Determine the
direction of the object's acceleration at a point
along the slide before it reaches its highest
point. The velocity at each point along the
slide is tangent to the slide. The speed of the
object decreases as it rolls up the slide.
Earlier Velocity: Draw an arrow
representing the velocity vo of the object
at time to when at a position just before
the desired point on the slide.
Later Velocity: Draw another arrow
representing the velocity vf of the object
at a later time tf when at a position just
after the desired point on the slide.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
- 10 -
©LC, tlo
Motion Diagram 2 - 5
An object is rotating in uniform circular
motion. Determine the direction of the
object's acceleration at a point A and B along
the circle. The velocity at each point along the
circle is
tangent to the circle. (You will need to do a
motion diagram 2 at position A and at position
B.)
Earlier Velocity: Draw an arrow
representing the velocity vo of the object
at time to when at a position just before
the desired point on the circle.
Later Velocity: Draw another arrow
representing the velocity vf of the object
at a later time tf when at a position just
after the desired point on the circle.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Motion Diagram 2 - 6
Revised 8/08
- 11 -
©LC, tlo
An object is rotating in elliptical motion.
Determine the direction of the object's
acceleration at a points A and B along the
ellipse. The velocity at each point along
the ellipse is tangent to the ellipse. (You will
need to do a motion diagram 2 at position A
and at position B.)
Earlier Velocity: Draw an arrow
representing the velocity vo of the object
at time to when at a position just before
the desired point on the ellipse.
Later Velocity: Draw another arrow
representing the velocity vf of the object
at a later time tf when at a position just
after the desired point on the ellipse.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Motion Diagram 2 - 7
Revised 8/08
- 12 -
©LC, tlo
Description:
Earlier Velocity: Draw an arrow
representing the velocity vo of the object
at time to when at a position just before
the desired point on the slide.
Later Velocity: Draw another arrow
representing the velocity vf of the object
at a later time tf when at a position just
after the desired point on the slide.
Velocity Change: Subtract the two
velocity vectors to find the change in
velocity v = vf - vo during the time
interval t = tf - to. To do this, place
the tails of vo and vf together. Then,
v is a vector that points from the head
of vo to the head of vf.
Acceleration: The acceleration equals
the velocity change v divided by the
time interval t needed for that change
-- that is, a = v/t . In this example
we do not know the time interval.
However, you can determine the direction
of the acceleration since it points in the
same direction as v.
Revised 8/08
- 13 -
©LC, tlo